openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Counting /CompanionInitialization.lean
| import OAI.Combinatorics.ContingencyTables.Counting.CompanionFactorialApplication | |
| import OAI.Combinatorics.ContingencyTables.Transport.InitialTransversals | |
| namespace OAI | |
| /-! # Exact initial normalizer for the literal companion completion count -/ | |
| namespace ContingencyTables.CompanionInitialization | |
| open scoped BigOperators Classical | |
| open CommonBasesFPRAS CompanionFactorialApplication LabelledFactorial TwoLiftStates | |
| variable {I J : Type*} [Fintype I] [Fintype J] | |
| noncomputable def freeTotal (A : Instance I J) : ℝ := | |
| ∏ a : ForestAssignmentSplit.FreeCell A.small A.T, | |
| ((A.b a.val.1 a.val.2+1:ℕ):ℝ) | |
| theorem h_zero (A : Instance I J) (z : View A → ℕ) : h A 0 z=freeTotal A := | |
| ForestCountIdentification.sourceMarginal_zero A.small A.T A.d A.b A.r A.c z | |
| theorem denominator_eq (A : Instance I J) : | |
| CompanionFactorialApplication.denominator A= | |
| InitialTransversals.denominator (fun s : SmallCell A => A.b s.val.1 s.val.2) := by | |
| unfold CompanionFactorialApplication.denominator TwoLiftStates.denominator InitialTransversals.denominator | |
| simp_rw [label_width] | |
| /-- C(0) is derived for the actual source f, using the finite | |
| paired-transversal enumeration and the literal free-completion count. -/ | |
| theorem total_zero (A : Instance I J) : | |
| slotTotal pairs (f A 0) ∅=freeTotal A*∏ s : SmallCell A, ((A.b s.val.1 s.val.2:ℝ)+1) := by | |
| have he : f A 0=firstLift (group (labelCell A)) (fun _ => freeTotal A) | |
| (CompanionFactorialApplication.denominator A) := by | |
| funext S | |
| unfold f firstLift | |
| rw [h_zero] | |
| rw [he,denominator_eq] | |
| convert InitialTransversals.slotTotal_constant | |
| (fun s : SmallCell A => A.b s.val.1 s.val.2) (freeTotal A) using 1 | |
| congr 1 | |
| end ContingencyTables.CompanionInitialization | |
| end OAI | |